Post History
#5: Post undeleted
by
JRN
·
2021-08-22T02:33:30Z (over 3 years ago)
#4: Post edited
by
JRN
·
2021-08-22T02:33:02Z (over 3 years ago)
Fixed error
- In equations 1.9 and 1.10, the quantities $\mathbf{r}$, $\mathbf{F}$, $\mathbf{N}$, and $\mathbf{v}$ are vectors, and the symbol $\times$ denotes a vector cross product operation.
- The [product rule](https://en.wikipedia.org/wiki/Cross_product#Differentiation) of differential calculus also applies to the vector cross product.
- That is, $\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{a}\times \mathbf{b})=\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{a})\times\mathbf{b}+\mathbf{a}\times\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{b})$.
- Thus,
- \begin{align*}
- \frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v}) &=\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r})\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
- &=\mathbf{v}\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
\end{align*}
- In equations 1.9 and 1.10, the quantities $\mathbf{r}$, $\mathbf{F}$, $\mathbf{N}$, and $\mathbf{v}$ are vectors, and the symbol $\times$ denotes a vector cross product operation.
- The [product rule](https://en.wikipedia.org/wiki/Cross_product#Differentiation) of differential calculus also applies to the vector cross product.
- That is, $\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{a}\times \mathbf{b})=\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{a})\times\mathbf{b}+\mathbf{a}\times\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{b})$.
- Thus,
- \begin{align*}
- \frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v}) &=\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r})\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
- &=\mathbf{v}\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
- \end{align*}
- Since $\mathbf{v}\times m\mathbf{v}=m(\mathbf{v}\times\mathbf{v})=m\mathbf{0}=\mathbf{0}$, we have
- \\[\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v})=\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\\]
#3: Post edited
by
JRN
·
2021-08-22T02:23:35Z (over 3 years ago)
Fixed error
In equations 1.9 and 1.10, the quantities $\mathbf{r}$, $\mathbf{F}$, $\mathbf{N}$, and $\mathbf{v}$ are vectors, and the symbol $\times$ denotes a vector cross product.I am assuming that the vector $\mathbf{r}$ is not a function of time but $\mathbf{v}$ is. If so, then $\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v})=\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})$.For some reason, the reference you are using wants to use the equation\\[\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v})=\mathbf{v}\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\\]But this is just- \begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v}) &=m(\mathbf{v}\times\mathbf{v})+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline&=m(\mathbf{0})+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})ewline&=\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})ewline- \end{align*}
- In equations 1.9 and 1.10, the quantities $\mathbf{r}$, $\mathbf{F}$, $\mathbf{N}$, and $\mathbf{v}$ are vectors, and the symbol $\times$ denotes a vector cross product operation.
- The [product rule](https://en.wikipedia.org/wiki/Cross_product#Differentiation) of differential calculus also applies to the vector cross product.
- That is, $\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{a}\times \mathbf{b})=\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{a})\times\mathbf{b}+\mathbf{a}\times\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{b})$.
- Thus,
- \begin{align*}
- \frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v}) &=\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r})\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})
- ewline
- &=\mathbf{v}\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})
- ewline
- \end{align*}
#2: Post deleted
by
JRN
·
2021-08-22T02:13:19Z (over 3 years ago)
#1: Initial revision
by
JRN
·
2021-08-22T02:03:06Z (over 3 years ago)
In equations 1.9 and 1.10, the quantities $\mathbf{r}$, $\mathbf{F}$, $\mathbf{N}$, and $\mathbf{v}$ are vectors, and the symbol $\times$ denotes a vector cross product.
I am assuming that the vector $\mathbf{r}$ is not a function of time but $\mathbf{v}$ is. If so, then $\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v})=\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})$.
For some reason, the reference you are using wants to use the equation
\\[\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v})=\mathbf{v}\times m\mathbf{v}+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\\]
But this is just
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}t}(\mathbf{r}\times m\mathbf{v}) &=m(\mathbf{v}\times\mathbf{v})+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
&=m(\mathbf{0})+\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
&=\mathbf{r}\times\frac{\mathrm{d}}{\mathrm{d}t}(m\mathbf{v})\newline
\end{align*}